Unsupported Precision Round Rail Calculator

[Ross77]

OK so I as a result of a question on another thread here is a very Basic calculator for precision round rail.

Unfortunately its not the 'bells n whistles' version I had planned but hopefully is OK as a start as time is in short supply at the moment. I'm hoping to add to it and make it a bit tidier as time is available.

It's only intended as a guide and you will have to make your own decision as to which answer you use, it can be used for calculating the deflection of a design or what the max load would be for a given bar, span and accepted deflection. Comments and abuse accepted and welcome:heehee:

Now the disclamer.........
Supplied in good faith and to the best of my knowledge is correct. blah blah... :smile:

[Wobblybootie]

Cheers ... Just what the doc ordered!!:beer::beer:

[routercnc]

Nice one Ross, looks good to me. I fed those numbers into my Gantry Spreadsheet (using tube profile with OD 16 and ID 0, length 900mm and load 250N) and got the same values for second moment of area and vertical deflection.

Out of interest, I've posted my sheet. It compares different sections and how they deflect due to vertical load and tool cutting applied torsion. Obviously, this spreadsheet only considers the gantry part of the machine.
Purple boxes are inputs, yellow boxes are outputs. There are graph tabs to visually compare sections.

Might help other people with the design process, although usual disclaimers apply...


EDIT: Updated to version 4, with much clearer layout and data entry. Also added different end conditions (from Ross's sheet).

[Ross77]

Thanks Barry, lots of equations to steal, Umm i mean use.....:whistling: thats a good idea to have them on one sheet and then linked to graphs. I like it. It wouldn't take to much to add the resonant and harmonic frequencies as well.


I fed those numbers into my Gantry Spreadsheet (using tube profile with OD 16 and ID 0, length 900mm and load 250N) and got the same values for second moment of area and vertical deflection.

Thats a relief, I'm good at finding fault with other peoples work but then make stupid mistakes in my own :whistling:

just a couple of points I'm unsure of.... It states simply supported for the vertical deflection but fixed for the torsion calcs. Are they separate calcs or are you assuming that the hinge joint can resist axial rotation?

Also I have been after an equation for the torsion constant (k) for a while but I always understood that it is pain to calculate for anything other than a cylinder. The program I have gave a different answer than you for the 80 x 80 x 3mm box section. I make it 1.37 e6!! [1370000] (not had chance to fire up the FEA yet so it might be the free programs I'm using, but it has always been correct before).

I only ask because I've been trying to think of a way to do what you have done for a while now but have been stuck at the torsion bit and have to resort to individual section checking of FEA. I think the only real solution is to create (or find) a data base of the most used sections (inc extruded t slot) and put them on a pick list in excel

[Ross77]

The program I have gave a different answer than you for the 80 x 80 x 3mm box section. I make it 1.37 e6!! [1370000]

Ok found it. It looks like a 'times by 4' has crept in at the begining of the equation. If you remove it the answer becomes 1.37e6 and the torsional resistance goes up on the graph :smile:

Do you mind if I add to it or is a work in progress yourself?

Cheers
Ross

[routercnc]

Hi Ross,

Yes, I'm assuming the hinge can resist torsion. I'm trying to ignore the rest of the machine and just compare the sections themselves to show the differences.
For bending the load case is simply supported (i.e hinges), with load in the centre.
For torsion the ends are constrained, and the load is applied in the centre of the span, hence the x4 factor.

You might have made the same mistake I did initially (or may be I am wrong !), which is to find out the stiffness of a beam with just one end restrained and the other far end twisted. This is not the case with a gantry, where both ends are restrained and the torsion is applied in the centre. I think these two factors will give a x4 stiffness increase. Your FEA may be for a simple beam with one end constrained and the other far end twisted. Let me know if you follow this argument! I guess they are both 'right' in their own way.

Nice to have a double check on the torsion anyway. I think the x4 is just the end condition and twisting point difference.

It would be good for Lee to collect this sort of information together and post it somewhere appropriate on the site. I'm very happy for you (or others) to modify this sheet, but as it works well at the moment would prefer to see any mods as a seperate posting (rather than overwritten).

I use Roark's Formulas for Stress and Strains, which is a weighty book and takes a long time to wade through and get what you want, but it is all in there. Turning it into Excel versions with graphs helps me visualise the effects of different parameters.

I've been working on another spreadsheet which analyses the principles involved in cnc router (or mill) design, and the effects of geometry and inertia. When I feel ready I'll post this as well . . .

Barry

[Ross77]

Hi Barry.

Ok I get it now, you have just short circuited the calc and mulitplied the constant by 4. Is that from Roarks or an asumption?


You might have made the same mistake I did initially (or may be I am wrong !), which is to find out the stiffness of a beam with just one end restrained and the other far end twisted.

I got confused because the torsion constant is a function of its shape and not the loading condition (being a constant and all :smile:) but thats the joy of engineering eh ? lots of ways to acheive the same result.

I didnt mean I wanted to overwrite your sheet just asking for permission to use elements of it in another form.


I've been working on another spreadsheet which analyses the principles involved in cnc router (or mill) design, and the effects of geometry and inertia. When I feel ready I'll post this as well . . .


Looking forward to seeing that.

[routercnc]

Hi Ross,

You're right. I just stuck the x4 in somewhere convenient - I only wanted the tool deflection value out at the end, which I guess is what most people might want for design work. If you want the torsion constant out as well then this x4 should go somewhere else in the calculation chain. One option would be to divide the applied torque by 4.

The x4 is an assumption based on reasoning (it's probably in Roark's somewhere but it is quite a read), but I ran it past a colleague and he came to the same conclusion. A bar with a twisting load applied at one end and the other fixed, will be twice as stiff if the load is then applied to the 'centre' of the span (x2). If the free end is then also restrained, with the twisting load still at the centre, then this previously free part of the bar will now also be able to resist the twist, doubling the stiffness again (x2). This is a total of x4. Technically the spreadsheet should be modified to be more correct (in the workings) although the end results are OK. Do you agree with this logic?

If you want to post a modified sheet, or use the equations in your own sheet then go for it. It would be great if there was an special design/calculation area on the forum where people who were asking "what sort of section is best for . . " or "how much deflection might I have . . ." could download a simple spreadsheet and get a rough idea. I think Irving2008's motor calculation was a great example of this.

Barry

[Ross77]

You're right. I just stuck the x4 in somewhere convenient


No problem with that. Thats what its all about re-engineeering equations to suit the problem in hand but it is goood practice to state any changes and the reasoning behind them. stops people like me asking questions :smile:

I'm sure all this could be resolved with a good fashion conversation. pm me your no. or maybe we will have to try the skype option....?

[Normsthename]

I tried opening it, but my Excel won't open it for some reason :mad:
Could someone do me a quick calculation
I am upgrading my Homebuilt Plasma by extending the table, this is the design so far :smile:
The Y Axis has two unsupported solid rails @1500mm long (See Picture) the centre bar is the Ballscrew.
I still need to add some form of aluminium extrusion to stiffen up the Y Axis Gantry.
It has the Z Axis assembly hanging off of it and also the Plasma torch, not sure what that will weigh.......
I think I might need to put some form of support under the rail, I had to do that on my first machine, I had small angles welded at 45° under the rails, and setscrews to bear onto the rail

Thanks in advance :smile:

Andy

[Ross77]

Hi Andy.

Try this version, I have back saved it in compatibility mode. It is best if you have a play with the figures as it with give an idea how the weight and length affect the deflection. and it might scare you.....:twisted:

You didn't state a bar size so a quick calc using 25mm dia. 1500mm span and 250N (25 kg) load the best case deflection would be just over a mm but worst case is 5mm depending on the restraints.

You really need additional cross rails to support the gantry sides as at the moment the rails are taking all the loads as well or you will have to assume simply supported. :eek:

I think the biggest problem you will have is the bounce caused by vibration. but that can be sorted by not letting the torch travel at resonant frequencies and keeping the acceleration and deceleration down.

Let know if you can open that one either.

[Normsthename]

Thanks Ross, that version loads fine :smile:
I have added a couple of lengths of 40mm aluminium extrusion to the model

2360
I have one small problem of the unsupported rails being 30mm too short :redface:
I may need to move the X Axis rails on top of the box section to give me a bit more room to play with :smile:


I think the biggest problem you will have is the bounce caused by vibration. but that can be sorted by not letting the torch travel at resonant frequencies and keeping the acceleration and deceleration down.
One thing I want from the new machine is to keep the speed up on corners etc
My current machine has a really heavy steel gantry (I originally built is as a Router), and I have to keep the acceleration and speeds down on corners.
On straight sections it cuts parts to almost LASER Quality, when the machine has to slow down for corners it causes dross to build up on the corners :cry:

Andy

[routercnc]

I've updated the previous gantry stiffness spreadsheet with a few minor tweaks (one slight error corrected, factor 4 re-located). New version 3 below:

This will calculate the vertical deflection of various types of section, plus twisting and resultant tool deflection due to cutting forces (on routers etc.)

Instructions for use
1. Top left boxes:
Load applied to section
Cutting force in that material (see sheet for suggested values - thank you Irving2008 for these values, borrowed from stepper motor calc sheet!)
Length of beam

2. Next set of top left boxes are for the material.
Defaults are steel, but there is a lookup table off to the left for aluminium or wood (Douglas Fir). Copy over the 3 relevant values into Young's modulus, Density, and Modulus of rigidity.

3. Geometry
The shape inputs are next in rows 15 to 17. Purple boxes.

4. Results
The yellow boxes in row 25 and below show various outputs.

Enjoy !

[routercnc]

Hi Andy,

Your machine is starting to look really good. A suggestion (from a non-plasma expert!) is to consider running a simple bearing from the back of the Z axis plate onto the top and bottom of the upper 40mm rail. This would give a bit more support to reduce the sag, and possibly help with the resonance (if that is due to the round unsupported rails vibration). Just a thought . . .

In terms of running the x axis rails on top, rather than on the sides, I've always wondered if this was easier from a build point of view. As drawn the gantry has to be very precisely made so that it is a nice fit onto the rails. The rails or gantry might need to be shimmed to achieve this. If you put the rails on top then everything can be bolted together and settled before finally marking the holes into the frame. Maybe there is another way but this seems a bit easier to me.

Good luck with it,
Barry

[Ross77]

Glad it works and is of some use to someone :smile:

Your machine is looking good. what CAD package is that?

With regard the rail orientation then I don't think it really matters for a plasma machine. Mounting them on the side will make the most of their strength and equal out the load capabilities but as you are only moving a light gantry then you shouldn't be getting any where near the max strength of the bearings. There is also the point that the cheaper open bearings can bend and open up even more if mounted on their side.

I've not had anything to do with plasma or even looked into it but it seems to me that if the z axis only has to hold the weight of the torch then there is scope to really reduce the weight of it. Which will help with the deflection and vibration.

I would have thought that you could use sheet metal instead of the plate, get the smallest dia ballscrew (or use belts) and 8mm round rails. Really lean it out.

Is that a double travel arrangement that you have on the z axis, again not a plasma expert but it looks like its just adding extra weight :smile:

[Ross77]

A suggestion (from a non-plasma expert!) is to consider running a simple bearing from the back of the Z axis plate onto the top and bottom of the upper 40mm rail. This would give a bit more support to reduce the sag


While this seems to make good sense it would appear to me to transmit any imperfections in the ali section to the rail and carriage. If you go down this route then there is no reason to use the precision rail and and I would just mount more bearings and run it on the Ali sections full stop.

Barry, I see that is how you have done it on the x axis on your machine, do you get any problems with surface Finnish or the bearings compacting the Ali and causing a groove?

[Normsthename]

what CAD package is that?
It was drawn on Solidworks :smile:
I am lucky in having use of Solidworks thru' the Company that I do work for
It is a brilliant package for designing CNC Tables, you can animate all the gantry movements etc
The picture I posted was created after I loaded the assembly into Photoview that creates a 'Rendered' view of the model
This is what the model looks like when actually designing it.

2363

The weldments tool is brilliant, I drawn the complete framework for the table in 20 minutes :smile:


Your machine is starting to look really good. A suggestion (from a non-plasma expert!) is to consider running a simple bearing from the back of the Z axis plate onto the top and bottom of the upper 40mm rail. This would give a bit more support to reduce the sag, and possibly help with the resonance (if that is due to the round unsupported rails vibration). Just a thought . . .
Not a bad idea :smile:

Is that a double travel arrangement that you have on the z axis, again not a plasma expert but it looks like its just adding extra weight
That is a vital part of a Plasma system :smile:
It is a 'Floating switch'
What happens is Z axis lifts up and moves to its pierce position, the Z axis then lowers until the Torch makes contact with the material.
The torch is fitted to the second linear slide, and just rests using gravity.
When the torch touches the material it trips a micro-switch, this then sends a signal to Mach3 and zeros the Z DRO
I will start a new build thread for my new machine, keep the suggestions coming! :smile:

Andy

[routercnc]

Hi Andy,
I was also wondering about the double Z axis arrangement, thanks for explaining.

Ross,
Yes, there are some risks from bearings on ali section. My machine is still some way from running yet so I don't know how it will cope with bearing surface wear. I picked some reasonably sized bearings to reduce the surface stress. The bearings are on axles which can be adjusted in slots, so I can add a running surface at a later date.
Since I'm only machining balsa and liteply, and all 2D (formers, ribs etc.) , my requirements are more simple than some and I think this will be OK for my needs. I know what I'd really like to do, but it's all a question of finance !

Barry

[Ross77]

It was drawn on Solidworks :smile:
Thought it looked expensive....


That is a vital part of a Plasma system :smile:
It is a 'Floating switch'
What happens is Z axis lifts up and moves to its pierce position, the Z axis then lowers until the Torch makes contact with the material.

I see, still seems a bit overkill :heehee:, couldnt you just use a circular spring loaded mount on the torch?


Ross,
Yes, there are some risks from bearings on ali section. My machine is still some way from running yet so I don't know how it will cope with bearing surface wear.

I will be inrested to know how you get on as the ali beam i just bought uses a similar idea but with rubber bearings and I was wondering how well it would work if I replaced them bearings. :smile:

[routercnc]

Updated stiffness spreadsheet to version4, see post #3 in this thread.

Now has much clearer layout, drop downs for some selections, and added Ross's end conditions for bending load case.

[Ross77]

That looks better Barry (not that there was anything wrong with the first one)

One point Ive just realized for both our sheets is that it only deals with one beam or bar when in reality two will be used, :eek: especially for a gantry!

The deflection is easy as you just halve it, but could be a bit more difficult for the torsion calcs. Rotation about the central point with torsion and horizontal deflection? nasty.

[routercnc]

Prompted by a recent post asking about deflection of profile sections layed out in an 'L' shape, here is a spreadsheet to calculate I values for almost arbitrary shapes.

These I values can then be used to calculate deflections. I'm in the process of adding this to the earlier spreadsheet, plus a few more new features, and hope to have something out early next week. For now, here is this mini calculator on it's own:

[Tom]

Nice one Barry & Ross, I've just been through these sheets... This sort of stuff only usually gets done if you're spending someone else's money :)
I wish I'd run calcs like this on my Y beam before building. I don't really need to now - I already know the answer :-~

Nice work,,,,

[Jonathan]

Those sheets are great!

I've just found that SolidWorks calculates moments of inertia, including second moment of area. I just drew out the SBR20 rail (20mm, supported) to try and compare it to unsupported rail. Here's the results for just the aluminium support on the rail:



Section properties of the selected face of SBR-20 supported round rail

Area = 400.80 millimeters^2

Centroid relative to assembly origin: ( millimeters )
X = 22.50
Y = 5.85
Z = 0.00

Moments of inertia of the area, at the centroid: ( millimeters ^ 4 )
Lxx = 8228.45 Lxy = 0.00 Lxz = 0.00
Lyx = 0.00 Lyy = 41748.33 Lyz = 0.00
Lzx = 0.00 Lzy = 0.00 Lzz = 49976.78

X is along the width of the rail, Y is the height and Z is the length.
I for the 20mm steel bar is 7840mm^4 ... so how do I combine these two values? Solid edge will calculate them as one but of course doesn't take into account the different moduli(?) for aluminium and steel.

EDIT: I've just found out how to do it - 'Parallel axis theorem'. Why can't it be simple!

[Jonathan]

Right I've worked out I for the 20mm supported rails in both directions.
In X direction I=53643mm^4
In Y direction I=22257mm^4

To account for the two materials used in the rails I've just scaled I for the bar by 67/200 so if you enter the above values and select steeel in the spreadsheets in this thread it should work.

For comparison the 20mm unsupported rail is 7854mm^4 - so it's roughly 2.8 to 6.8 times weaker than supported rail which is just 'floating'.

(I'm fairly confident that these values are correct, though I've literally only learnt about moments of inertia in the past 2 hours and it's like 1am so no guarantees!)

[routercnc]

Hi Jonathan,

Glad to see the spreadsheet being used. I think they could still do with making even more user friendly. They're ok if you are used to doing this stuff and just want to save doing it by hand, but if this is new territory then you could easily be put off, or worse still, try to use them in the wrong way and get meaningless answers.

Your method of mixing materials by scaling Young's modulus has got me thinking . . . think it's OK.

Your estimate of stiffness increase for supported rails over unsupported (2.8 to 6.8 times better) neglects the part that the supported rail is bolted to (often a thick plate for Z axis), which is the main reason for doing it. You'll find if you factor in the mounting plate that the differences are significantly greater still. Good work though at 1 in the morning!

Other CAD s/ware, such as AutoCAD, CATIA will give properties including second moment of area, so for complex shapes this is another way to do it.

[Jonathan]

Your estimate of stiffness increase for supported rails over unsupported (2.8 to 6.8 times better) neglects the part that the supported rail is bolted to (often a thick plate for Z axis), which is the main reason for doing it.

Yes, that's what I meant by the rail being 'floating'. I'm thinking of mounting my Y-axis rails either on 50x50x3mm steel box section, or 80x80x3 aluminium so I'm going to calculate both to compare. At risk of straying off topic are the sides of box section parallel enough to mount rails to directly?

Can you link me to somewhere that details how to calculate the twisting?

Other CAD s/ware, such as AutoCAD, CATIA will give properties including second moment of area, so for complex shapes this is another way to do it.[/QUOTE]

[routercnc]

Hi Jonathan,

Ah sorry, missed what you meant by floating.

Ali vs steel parallel question. I've always thought that items like ali profile or aluminium extruded RHS etc. are pretty true due to the extrusion process used. Steel box can be resistance welded along one edge, which I would suspect is not so good. If going for steel sections on critical parts which held the rails then I'd be considering having the faces ground. But I'm not an expert to my suggestion is that you post this up for a better response!

Twisting calculations were from Roark's formulas for stress and strain, an old University text book I have about 1" thick full of tables and equations. Excellent book, but heavy going in places. If you search the internet for 'twisting constant K', which is a bit like I values for bending, you might get somewhere.

[Jonathan]

I've posted the parallel question here:

http://www.mycncuk.com/forums/showthread.php?2213-Straightness-of-steel-alumunium-box-section&p=15781#post15781



Twisting calculations were from Roark's formulas for stress and strain, an old University text book I have about 1" thick full of tables and equations. Excellent book, but heavy going in places. If you search the internet for 'twisting constant K', which is a bit like I values for bending, you might get somewhere.

Thanks, I'll look into it...

[Ross77]

I started off down that path of calculating the I values of each bit in Autocad and the parallel axis etc...... but gave up on the torsion constant for irregular shapes, when I was looking on the net I kept seeing reference to to the 2 polar moment which CAD can also work out but this is only true for circular tube and bar. I only gave up because I can now use FEA to do it all for me. if you get any where then I sure would like to know.

I also found that the more you analise it the more time it takes and you soon realise that, as Barry pointed out, just doing a few basic cals on the main structure will be sufficient. adding the rails etc becomes the factor of safety that seems to be missing from a lot of calcs. :naughty:

[Jonathan]

... I can now use FEA to do it all for me. if you get any where then I sure would like to know.

I also found that the more you analise it the more time it takes and you soon realise that, as Barry pointed out, just doing a few basic cals on the main structure will be sufficient. adding the rails etc becomes the factor of safety that seems to be missing from a lot of calcs. :naughty:

I've certainly spent a lot of time analysing it now and have to say I'm still not happy. I think I tend to agree now about just doing basic calcs.

Can you point me to any FEA program I can get hold of? I was actually going to ask this in my last post!

[Ross77]

for FEA i use AutoCAD inventor (Student License) because I thought it would be an easy transition from Normal AutoCAD, not the case tho, apparently its more like Solidworks where you work parametrically. Has Soildworks not got a FEA module I'm sure Ive seen them generated from solidworks.

Its worth down loading the trial versions, if you already have the 3d model then at least theres 30 days of free analysis :whistling: